Question: Ben is 5 times as old as Daniel. Six years ago, Ben was 7 times as old as Daniel. How old is Ben now?
Solution: We can use the given information to write down two equations that describe the ages of Ben and Daniel. Let Ben's current age be $b$ and Daniel's current age be $d$ The information in the first sentence can be expressed in the following equation: $b = 5d$ Six years ago, Ben was $b - 6$ years old, and Daniel was $d - 6$ years old. The information in the second sentence can be expressed in the following equation: $b - 6 = 7(d - 6)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $b$ , it might be easiest to solve our first equation for $d$ and substitute it into our second equation. Solving our first equation for $d$ , we get: $d = b / 5$ . Substituting this into our second equation, we get: $b - 6 = 7($ $(b / 5)$ $- 6)$ which combines the information about $b$ from both of our original equations. Simplifying the right side of this equation, we get: $b - 6 = \dfrac{7}{5} b - 42$ Solving for $b$ , we get: $\dfrac{2}{5} b = 36$ $b = \dfrac{5}{2} \cdot 36 = 90$.